We cover recent developments in conformal geometry, mainly on 4-manifolds, including 1) study of integral conformal invariants, /sigma_2 functional on 4-manifolds, issues related to compactness, and study of blow-up limits. (Recent joint work of Chang-Yang-Ruobing Zhang), and 2) study of conformal 'filling in' of manifolds as boundary of conformal compact Einstein manifolds, the questions of existence and non-existence; unique and non-uniqueness and compactness with given boundary data, including recent developments in this field by Qing-Li-Shi, Gursky-Han, Gursky-Han-Strolz, Chang-Ge and Chang-Ge-Qing.